I have heard that the working solution is only good for a few hours. If that is the case, how long would the stock solution last?

Right. Pure solutions of sodium sulfite are not stable, the effectiveness (as antioxidanz) halves itself after approximately one week. Solutions of developer and sulfite lean mutually. For the use as HCA, this quality is not important. Selfmade HCA is rather cheap, more than 20-30 11x14 prints per day is rare to do, but with less than ten prints, the capacity of one liter is not exhausted. The solution could be used without disadvantages on the following day.
The concentrate will last some weeks or more. The Bisulfit stabilizes, the acid additions lower the pH level. The exchange of hypo against sulfite also works with older (stock) solution.